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Geometry of differential operators on Weyl manifolds. (English) Zbl 0906.53034

A Codazzi structure on an \(n\)-dimensional manifold \(M\) is a pair \(({\mathcal C}, {\mathcal P})\) where \({\mathcal C}= [g]\) is a conformal class of semi-Riemannian metrics and \({\mathcal P}= [\nabla]\) is a projective class of torsion-free connections on \(M\), such that there exist a connection \(\nabla\) in \({\mathcal P}\) and a metric \(g\) in \({\mathcal C}\) which satisfy the Codazzi equation \((\nabla_Xg) (Y,Z)= (\nabla_Yg) (X,Z)\) for all vector fields \(X,Y,Z\) on M. A Codazzi manifold is a manifold endowed with a Codazzi structure. On the other hand, a Weyl structure on \(M\) is a pair \(({\mathcal C}, \nabla)\) where \(\nabla\) is a torsion-free connection on \(M\) such that for each metric \(g\) in the conformal class \({\mathcal C}\) there exists an associated 1-form \(\omega\) which satisfies \(\nabla g= \omega \otimes g\). A Weyl manifold is a manifold endowed with a Weyl structure.
In this paper, first of all, the close relations between Codazzi and Weyl structures is studied. So, it is shown that the set of all Codazzi structures giving rise to a given Weyl structure is parametrized by the apolar (1,2) tensor fields.
A second order partial differential operator \(D\) on a manifold with a conformal class of semi-Riemannian metrics \({\mathcal C} =[g]\) is said to be of Laplace type if the leading symbol of \(D\) is given by a metric \(g\) in \({\mathcal C}\). For any positive smooth function \(\beta\) on \(M\), denote by \({\mathcal M} (\beta)\) the corresponding multiplication function and by \(_\beta D\) the operator obtained from \(D\) by using the conformal change \(g\to \beta g\). \(D\) is said to transform conformally if \(_\beta D= {\mathcal M} (\beta^a) \circ D\circ {\mathcal M} (\beta^b)\) for \(a+b =-1\).
Several natural operators of Laplace type are considered; in particular, the conformal Laplacian, which is an example of such an operator, is generalized to Weyl and Codazzi geometry. In the Riemannian case, i.e., when \(g\) is a positive definite metric, as it was obtained by the second author [P. B. Gilkey, ‘Invariance theory, the heat equation and the Atiyah-Singer index theorem’, 2nd. ed. (Stud. Adv. Math., CRC Press, Boca Raton/FL) (1995; Zbl 0856.58001)], there is an asymptotic expansion as \(t\downarrow 0\) of \(\text{Tr}_{L^2} e^{-tD}\) which gives invariants \(a_m(D)\). It is also known that if \(D\) transforms conformally, then \(a_m(D)= a_m(_\beta D)\) holds from the confrmal index theorem.
The last section of this paper is devoted to obtain several results which can be used to extend the invariants \(a_m\) from the Riemannian to the semi-Riemannian category.
Reviewer: A.Romero (Granada)

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53A30 Conformal differential geometry (MSC2010)

Citations:

Zbl 0856.58001
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